low power and high-speed implementation of fir …

Revised Paper: ToW 03-582

FIR Filter Implementation for SDR 1

LOW POWER AND HIGH-SPEED IMPLEMENTATION OF FIR FILTERS FOR SOFTWARE DEFINED RADIO RECEIVERS

A. P. Vinod and E. M-K. Lai

School of Computer Engineering, Nanyang Technological University Singapore 639798

Tel: (+65) 67906258 Fax: (+65) 67926559 Email: {asvinod, asmklai}@ntu.edu.sg

Abstract

The most computationally intensive part of the wideband receiver of a software defined

radio (SDR) is the intermediate frequency (IF) processing block. Digital filtering is the

main task in IF processing. The computational complexity of finite impulse response

(FIR) filters used in the IF processing block is dominated by the number of adders

(subtractors) employed in the multipliers. This paper presents a method to implement

FIR filters for SDR receivers using minimum number of adders. We use an arithmetic

scheme, known as pseudo floating-point (PFP) representation to encode the filter

coefficients. By employing a span reduction technique, we show that the filter

coefficients can be coded using considerably fewer bits than conventional 24-bit and 16-

bit fixed-point filters. Simulation results show that the magnitude responses of the filters

coded in PFP meet the attenuation requirements of wireless communication standard

specifications. The proposed method offers average reductions of 40% in the number of

adders and 80% in the number of full adders needed for the coefficient multipliers over

conventional FIR filter implementation methods.

Index Terms – IF processing, Pseudo floating-point representation, software defined

radio, FIR filter.



I. Introduction

SDR is fast becoming a crucial element of wireless technology. The use of SDR

technology is predicted to replace many of the traditional methods of implementing

transmitters and receivers while offering a wide range of advantages including

adaptability, reconfigurability, and multifunctionality encompassing modes of operation,

radio frequency bands, air interfaces, and waveforms [1]. Research in this field is mainly

directed towards improving the architecture and the computational efficiency of SDR

systems.

The most computationally intensive part of an SDR receiver is the channelizer since it

operates at the highest sampling rate [2]. Channelization in SDR receivers involves the

extraction of multiple narrowband channels from a wideband signal using several

bandpass filters called channel filters [3]. Low power and high-speed FIR filters are

required in the channelizer [4]. The key functional units in a digital filter are delay, adder,

and multiplier – out of which multiplier dominates the hardware complexity. It is well

known that by representing filter coefficients as sum-of-powers-of-two (SOPOT), each

multiplication in filtering can be replaced with simple shift-and-add operations [5]-[8].

The complexity of the FIR multiplier is dominated by the number of adders (subtractors)

employed in the coefficient multipliers. The number of adders needed in the multipliers is

proportional to the coefficient wordlength [9]. The fixed-point arithmetic implementation

of channel filters in digital wideband receivers requires 24-bit wordlength to meet the

channel specifications [10, 11]. It has been reported in [10] that the 16-bit filter

implementation results in a significant degradation in stop-band attenuation, failing to



meet the spectral mask requirements. The filters implemented using 24-bit and 16-bit

SOPOT coefficients require considerably larger number of adders and hence further

hardware optimization is required to meet the constraints of power consumption and

speed in SDR receivers [11]. In this paper, we present the implementation of FIR filters

using an arithmetic scheme called Pseudo Floating-Point (PFP) representation [12]. We

show that the coefficients can be coded using considerably fewer bits than the

conventional 24-bit and 16-bit implementations. The magnitude responses of the

resultant filters meet the spectral mask characteristic of the relevant standard for mobile

communications receivers. The contributions of this paper can be summarized as follows:

1. An efficient coefficient coding scheme using PFP representation for implementing FIR

filters in SDR receivers is proposed. By employing a span reduction technique, it is

shown that the wordlength of the filters can be minimized to 10 bits or fewer.

2. All previous work on filter implementation [5]-[9] discussed hardware reduction in

terms of the number of adders and has not addressed the complexity of adders. The

complexity of each adder employed in multiplication is significant for low power and

high-speed implementations. We analyze the complexity of implementation in terms of

full adders required for each multiplier of the filter. A low power, high-speed

implementation of PFP coded filters with a minimum number of full adders is proposed.

This paper is organized as follows: In section II, a brief review of the PFP representation

is provided. The PFP coding scheme for implementation of FIR filters is discussed in

section III. In section IV, a low power implementation of PFP coded filters is presented.



The design examples of FIR filters for the SDR receiver are illustrated in section V.

Section VI provides our conclusions.

II. The Pseudo Floating-Point Representation

The general representation of sum-of-powers-of-two (SOPOT) terms [5] for the ith filter

coefficient is

∑=−

=

1

02

B

j

ai

ijh . (1)

where B is the number of digits in the power-of-two representation. The expression for ih

can be rewritten as

∑=∑=−

=

−

=

− 1

0

1

02.2 2.2 000

B

j

caB

j

aaai

ijiiijih . (2)

where .0iijij aac −= The term 0ia is known as the shift and the upper limit value,

)( 0)1( iBi aa −− , is known as the span [12]. The bracketed term is known as the

normalised value ( n value). The shift and the normalised value are analogous to the

exponent and mantissa in true floating-point representations. Instead of expressing the

coefficients as a 16-bit integer, it can be expressed as a (shift, n-value) pair – this is the

definition of the pseudo floating point representation [12]. For a given coefficient set of

an N-tap filter, let L and M be the number of bits needed to encode the shift and n-value

respectively. Then,

)( max10

iNi

hshiftL−≤≤

= (3)

)( max10

iNi

hspanM−≤≤

= . (4)

The following example illustrates this concept. Consider the coefficient ),(nh whose 16-



bit SOPOT representation is given by .2222)( 14986 −−−− +++=nh This can be written

as ).2222(2 83206 −−−− +++ In this expression, the term 62− is the shift part (implying

‘right shift by 6’), and the bracketed term is the span part. The shifts are less complex

since they can be hardwired. Therefore, only 3 bits are needed for storing the shift value

(SOPOT representation of 6 is 110) and correspondingly, .3=L The span value, ,8=M

is obtained from the bracketed term. Hence the coefficient can be represented in PFP

using 11=+ ML bits, whereas its SOPOT representation requires 16 bits. In the case of

the practical filter implementation in [10, 11], the L and M values of 24-bit fixed-point

coefficients are 5 and 23 respectively. Hence, 28 bits are needed by the PFP for general

coefficient sets. For the 16-bit coefficients, L and M are 4 and 15 respectively and thus

require a total of 19 bits in PFP representation based on the examples in [10, 11]. It

would seem that the PFP representation might not be an optimal representation. However,

it would be interesting to investigate if the actual coefficient sets would require less than

the 28 bits and 19 bits in these cases. The span contributes significantly more to the

wordlength requirement than the shift. The shift values depend on the coefficient

wordlength and its maximum value is fixed based on the worst-case coefficient

(coefficient that has the largest power of two term) and so is not a parameter that could be

optimized further. Therefore, it is beneficial to explore some efficient means of reducing

the span without considerable implication on the magnitude response of the filter.

III. PFP Coding Scheme for FIR Filters

In this section, we show that by employing a span reduction technique, the wordlength

requirement of the FIR filters for an SDR receiver can be significantly reduced.



A. Span Reduction Technique

In our attempt to achieve a minimum wordlength for any coefficient set, we fix the shift

to the maximum value, l, corresponding to the worst-case coefficient set using equation

(3). The span value is progressively reduced by discarding the power-of-two terms and

checking whether the resulting filter response meets the filter specifications at each stage.

We can expect distortion in the frequency response characteristics, when such a span

reduction technique is employed to all the offending coefficients (offending power-of-

two terms here being defined as the power-of-two terms that exceed the lower bound

span). Our observation in employing the span reduction technique is that the pass-band

response of the resulting filter does not change drastically. It has also been noted that the

effect of span reduction on stop-band attenuation and peak stop-band ripple is minimal in

the case of filters having relatively few taps (filter length, ). 40<N The reason for this

behaviour can be explained as follows. Let the span value after performing the reduction

be .ms

1. In the case of lower order filters, the spans are closely distributed around ,ms whereas

for higher order filters the spans are widely distributed. Hence, the magnitudes of those

terms whose span exceed ,ms which are discarded are considerably smaller for lower

order filters when compared to that of higher order filters. As a result, the sensitivity of

the PFP coefficients to span reduction is very low. Sensitivity is a measure of the degree

of influence on the frequency response of a filter when any one of the coefficients is

quantized. The sensitivity )(ns can be computed by setting each coefficient, in turn, to

its nearest power of two, yielding in each case a response ),( iqH ω which gives:



∑ −==

M

iiiq HH

Mns

1

2)]()([1)( ωω . (5)

where )( iH ω and )( iqH ω are the frequency responses of the infinite-precision and

quantized coefficients respectively at M finite number of frequencies iω [13]. The

equivalent time-domain expression for sensitivity of an N-tap filter is given by

∑ −=−

=

1

0

2)]()([1)(N

nq nhnh

Mns . (6)

where )(nhq and )(nh represent the impulse responses of the quantized and infinite-

precision coefficients respectively. In the case of lower order filters, )]()([ nhnhq − is

small due to the distribution of spans close to .ms Hence, the sensitivity, which is a

square function is minimal and therefore the frequency response of the filters are almost

unaltered by the proposed span reduction technique. This will be illustrated in the design

examples provided in section V.

2. The span deviation from ms is relatively uniform across the different coefficients in

the case of filters with fewer taps when compared to that with larger taps. We have

investigated several examples of raised cosine filters with up to 40 taps corresponding to

different stop-band attenuation specifications. The floating-point filter coefficients were

generated using the “firrcos” function in MATLAB. Filter coefficients represented in 16-

bit SOPOT and 8-bit PFP forms were examined. Fig. 1 shows the average span

distribution across the coefficients of fifteen filters that we designed. These filters have

identical number of taps (i.e., 40) but different cutoff frequencies. Note that one half of

the symmetric filter coefficients are shown. The lower bound span value obtained using



the span reduction algorithm indicated by the horizontal dotted line in Fig. 1 is 5 bits. It

can be noted that the deviation of the spans of the coefficients )0(h to )18(h from the

lower bound value is uniform (within the rage of 5 to 7 bits) across the coefficient grid.

As a result, applying span reduction is similar to scaling the coefficients by the power-of-

two terms exceeding the lower bound span value. Scaling the coefficient set will not

affect the frequency response shape; instead it only changes the filter gain. In the case of

PFP representation for short filters, the change in gain is minimal since the span deviation

from ms is minimal. The worst-case span deviation occurs for the larger valued

coefficient ),19(h whose span is 9 bits. However, as it will be illustrated in the design

examples in section V, the magnitude of the offending power-of-two terms of the larger

valued coefficient is extremely small ( 142− to ).2 16− Hence the response deterioration

meets the stop-band attenuation specification when the PFP span reduction technique is

employed. It is worth to note that the tolerance scheme of Nyquist filters does not have a

constant maximum filter transfer function deviation with respect to the perfect Nyquist

characteristic. Instead, the tolerance scheme becomes wider towards the pass-band edge

[14]. Therefore, a minimal deviation of frequency response characteristics from the ideal

characteristic is acceptable in Nyquist filters.

B. PFP Coefficient Coding Algorithm

The steps for PFP coding of filters using the span-reduction approach are given below.

Step 1: Design the FIR filter, ),(nh as in the specification. Determine the frequency

response of the floating-point coefficients, ∑=−

=

−1

0.)()(

N

n

njd enhH ωω



Step 2: Set the coefficients to their closest SOPOT form using specified wordlength and

represent them as a (shift, span) pair in PFP. Fix the shift to the maximum value l,

corresponding to the worst-case coefficient set. Find the maximum span value, M. Set

iteration index to .0=k

Step 3: Decrease the span to M-1 by discarding the power-of-two terms of offending

coefficients and obtain the new set of coefficients, ).(nhq Determine ).()( nhnhq −

Step 4: The frequency response of the quantized filter whose span is reduced to M-1 can

be obtained using:

)()()( ωωω eidiq HHH += . (7)

In the time-domain, (7) can be expressed as

njN

nq

N

nienhnhnh ω−−

=

−

=∑ −+∑ )]()()([

1

0

1

0. (8)

where iω represents frequency samples in the stop-band. Obtain the frequency response

using equation (8).

Step 5: If ,)()( isiqs HH ωω ≤ where )( iqsH ω represents the stop-band response of the

PFP filter and )( isH ω as in the stop-band specification of the filter, set 1+= kk and go

to step 3. Otherwise, terminate the program and choose the PFP coefficients,

),(nhq corresponding to the ‘k’th iteration.



IV. Low Power Implementation

In this section, we present a method to implement the PFP coded filters with low power

and high-speed. Note that a reduced numbers of bits are required to encode the

coefficients employing the span reduction method and hence the filter can be

implemented using a minimum number of adders. Although minimizing the number of

adders reduces the complexity, it is also necessary to address the complexity of each

adder, which is significant for high-speed, low power implementations. An adder that

adds two n-bit numbers requires n full adders (FAs) to compute the sum. The area,

power, and speed of an adder depend on the adder width, n. Therefore, efforts to optimize

these parameters should focus on minimizing the adder width. We shall now obtain the

expressions for analyzing the complexity of each adder in the filter structure.

A. Adder Complexity

Definition 1 (Operands): The input signal shifted corresponding to the positional weights

of the nonzero digits in the coefficient form the operands of the adders. For example, in

the case of the coefficient, ,2222)( 14986 −−−− +++=nh if x represents the input

signal, the operands are ,6>>x ,8>>x ,9>>x and ,14>>x where ‘>>’ represents

shift right operation. The number of operands is equal to the number of nonzero digits in

the coefficient.

Definition 2 (Range): The range is defined as the number of bits of an operand. If x is an

8-bit signal, the ranges of the above-mentioned operands are 14, 16, 17, and 22

respectively. For an adder whose operands have ranges 1r and 2r such that ,12 rr > the

adder width is .2r Thus the adder would require 2r FAs to compute the sum.



Definition 3 (adder-step): One adder-step represents an adder in a maximal path of

decomposed multiplications. A multiplication can have different adder-steps, depending

on its structure. We employ the high-speed tree structure shown in Fig. 2, which uses the

minimum number of adder-steps. Coefficients with wordlengths up to 16-bits are

considered for analyzing the adder complexity and hence at the most sixteen nonzero

operands could occur in a multiplication.

Case I: Odd number of operands

Consider the filter tap shown in Fig. 3 that has an odd number of operands (nine), whose

ranges are indicated as .ir The sri shown adjacent to the adders represent the adder

widths. The total number of FAs required to implement this filter tap is

98642 32 rrrrrN FA ++++= . (9)

By extending this minimum adder-step structure to 16-bit coefficients, it can be shown

that the number of FAs, ),( oN required to compute the output of a filter tap with m (for

m odd) operands can be determined using the expression:

++++++++++++= 13111211910978756534312 222322 rarrarrarrarrarrarNo

151314131112119 3222 rarrarra ++++ . (10)

where ir is the range of the ith operand and the s'ia are equal to zero except for ,2−ma

which is 1. (Note that since we are considering coefficient wordlengths up to 16 bits, the

maximum possible value of m is 15). This can be illustrated using the example of the

filter tap ),(nho shown in Fig. 4. The coefficient ,22222 1514986 −−−−− ++++ is

considered where m is odd (five). The numerals adjacent to the data path represent the



number of bit-wise right shifts. If x is an 8-bit signal, the ranges, ,1r ,2r ,3r ,4r and ,5r

of the operands are 14, 16, 17, 22, and 23 respectively. Note that the adders ,1A ,2A ,3A

and 4A shown in Fig. 4 employed in multiplication have widths 16, 22, 22, and 23

respectively as indicated by the numerals in respective brackets. Using equation (10), the

number of FAs for computing )(ny is given by ,2 542 rrr ++ which is 83.

Case II: Even number of operands

Using the approach discussed above, the number of FAs, ),( eN required to compute the

output corresponding to a coefficient with m operands )16( ≤m is given by:

1614312210186042 432 rrcrcrcrrcrrNe +++++++= . (11)

where ,6m ,1

6for ,20

≠=

≡m

c , 10m ,1

10for ,21

≠=

≡m

c

10m ,2

12for ,32

≠=

≡m

c and 10m ,1

14for ,33

≠=

≡m

c .

For example, if six operands are present (i.e., ),6=m it would require )22( 642 rrr ++

FAs.

B. PFP Filter Implementation

In this section, we discuss the filter implementation using the proposed PFP coding and

compare it with the conventional SOPOT implementation. Consider the coefficient ),(nh

represented in SOPOT form, .2222 14986 −−−− +++ The output of the filter obtained

when )(nh is multiplied by x (8-bit signal) can be represented as



14986 >>+>>+>>+>> xxxx . (12)

In this case, there is an even number of operands (m=4) where 2r and 4r are 16 and 22

respectively. Using equation (11), the number of FAs for computing )(ny in SOPOT

implementation is 60. We shall now show that the number of FAs can be considerably

reduced using the PFP coefficients. The PFP representation of )(nh is

).2222(2 83206 −−−− +++ Fig. 5 shows the PFP implementation of this filter tap. The

ranges of the operands inside the bracket of )(nh (corresponding to its span bits) are 8,

10, 11 and 16. Thus, when compared with the conventional implementation, the adders

,1A ,2A and ,3A have smaller widths, since the ranges of their operands are smaller.

Using equation (11), the number of FAs for computing )2222( 8320 −−− +++x is 42.

The ‘shift right’ operation corresponding to the span )2( 6− of )(nh is performed after

the addition stages, as shown alongside the data paths at the output of adder .3A Note that

the shifts are hardwired and hence they do not have any cost implication other than a

minimal increase in chip area. The proposed PFP implementation requires only 42 FAs,

which is a reduction of 30% over the SOPOT implementation. The number of adder-steps

required to compute the partial products is two, which is the same as that of the

conventional method. The use of smaller adder widths in the PFP method reduces the

delay of each addition and hence the proposed method offers speed improvement over the

conventional method. By employing our span reduction algorithm, it is possible to

achieve substantially higher reductions of FAs.



V. Design Examples

Example 1: In this example, we use the SOPOT coefficients of the FIR subfilter in the

variable digital filter (VDF) based SRC (sampling rate converter) employed in the SDR

receiver presented in [15], to illustrate our method. The filter length chosen is 40 as in

[15]. The 20-bit SOPOT coefficients and the 10-bit PFP coefficients obtained after span

reduction are listed in Table I. Note that one half of the symmetric coefficient set is

considered. The sensitivity values of PFP coefficients obtained using equation (6) are

also shown in Table I. The shift value is fixed at 5=L bits based on the worst-case

coefficients, )1(h and )3(h (5 bits are required to store the maximum shift value, 16).

The lower bound of span )(M obtained by employing the proposed algorithm is 5 bits.

Hence the coefficient set is represented using 10 bits in PFP. The magnitude responses of

the filters are shown in Fig. 6. The red plot corresponds to the response of the filter

whose coefficients are coded in 10-bit PFP. The blue plot represents the response of the

conventional 20-bit SOPOT. Response of the 10-bit PFP filter shows close resemblance

to that of the 20-bit SOPOT filter. The peak stop-band ripple (PSR) of the PFP filter is –

24 dB and that of the SOPOT filter is –24.1 dB. Both filters offer identical peak pass-

band ripple (PPR) of 0.1 dB. These comparisons show that there is practically no

difference in the response of filters obtained using the proposed representation and the

conventional methods. The considerably low sensitivity values of PFP coefficients shown

in Table I account for this achievement. Fig. 7 shows the time domain characteristics of

the FIR filter implemented using the 20-bit SOPOT and 10-bit PFP versions of the

coefficients. The impulse response of the filter (symmetric half set coefficients) realized

using SOPOT coefficients (marked with blue colored star symbol) exactly coincides with



that of the PFP (marked with red colored square). It is observed from Fig. 7 that the zero

crossings of the impulse response remain unaltered when the PFP representation after

span reduction is used. Therefore, the FIR filters implemented using our method can

perform pulse shaping with minimal inter symbol interference (ISI).

Example 2: The W-CDMA receiver architecture presented in [16] is used to illustrate our

design in this example. The IF block of the receiver is shown in Fig. 8. The input

bandwidth of the IF signal covers one channel of 5 MHz in W-CDMA. The filter ),(1 zH

performs pulse shaping to achieve an attenuation of –40 dB at 5 MHz as in the W-CDMA

specification [17]. The output signal at 15.36 MHz, which is four times the W-CDMA

chip-rate of 3.84 Mc/s, is fed to base-band processing. The roll-off factor is selected as

0.22 for bandwidth efficiency in 3G cellular applications. A raised cosine filter of length

33 is designed. The lower bound PFP obtained is 8 bits. The PFP coefficients obtained

after span reduction and their sensitivity values with respect to 16-bit SOPOT coefficients

are listed in Table II. The magnitude responses of the filters are shown in Fig. 9. Both the

filters meet the desired attenuation of –40 dB at 5 MHz as in W-CDMA specifications.

The 8-bit PFP filter response shows close resemblance to that of the 16-bit SOPOT filter.

Table III shows the comparison of the numbers of adders required to implement the

filters in the design examples. The PFP implementation offers an average reduction of

44% over the conventional implementation in the design examples. Comparison of the

numbers FAs required to implement the multipliers for the filters using the SOPOT

method and the proposed PFP method are shown in Table IV. The percentage reductions

of FAs in the PFP method over the SOPOT method are also shown in Table IV. The



number of FAs shown in Table IV is computed as follows. Consider )4(h in Table I. The

16-bit SOPOT form of )4(h is .2222 19171310 −−−− ++− In this case, m is 4 and

considering an 8-bit signal, the values of 2r and 4r are 21 and 27 respectively. Using

equation (11), the number of FAs required to obtain the output of this filter tap is 75. The

PFP coding of )4(h is ).2222(2 973010 −−−− ++− The values of 2r and 4r are 11 and

17 respectively and only 45 FAs are required to obtain the output. (Note that the shift

operation ,2 10− can be hardwired). Thus, PFP coding before span reduction (BSR)

results in 40% reduction over 20-bit SOPOT coding. The 10-bit PFP form of )4(h

obtained after span reduction (ASR) is ).22(2 3010 −− − In this case, m is 2 and 2r is 11.

Therefore, only 11 FAs are required, which is a reduction of 85% over SOPOT coding. It

must be noted that this substantial reduction is achieved without any performance

deterioration on the frequency response of the filter.

VI. Conclusions

We have presented an efficient coefficient coding scheme using pseudo floating-point

representation and a span reduction algorithm for implementation of FIR filters in SDR

receivers. The computational complexity of the algorithm is relatively less since it is

applied only to the filter stop-band response samples. Our method can be used to

implement any FIR filters provided the number of taps is less than 40. A low power and

high-speed implementation using a minimum number of full adders is also proposed. Our

method offers average reductions of 40% in the number of adders and 80% in the number

of full adders over conventional FIR filter implementation methods.



Fig. 1. Average distribution of spans across the coefficient sets of 40-tap raised cosine filters.

Fig. 2. Tree structure employed for multiplication.

Fig. 3. Implementation of filter tap for odd number of operands.



)(nh 20-bit SOPOT Coefficients (shift: 5-bit, span: 5-bit)

10-bit PFP Coefficients Sensitivity ( 128=M )

)0(h 1715 22 −− + )22(2 2015 −− + 0

)1(h 2016 22 −− − )22(2 4016 −− − 0

)2(h 1812 22 −− + 122− 13101.1 −× )3(h 191716 222 −−− ++ )222(2 31016 −−− ++ 0

)4(h 19171310 2222 −−−− ++− )22(2 3010 −− − 13101.7 −× )5(h 18161511 2222 −−−− +++− )222(2 54011 −−− ++− 13101.1 −× )6(h 1817129 2222 −−−− −−−− )22(2 309 −− −− 12101 −× )7(h 20149 222 −−− ++ )22(2 509 −− + 15101.7 −× )8(h 1614118 2222 −−−− −−+ )22(2 308 −− + 11106.4 −× )9(h 19151298 22222 −−−−− −−−−− )222(2 4108 −−− −−− 12102.8 −× )10(h 1916131198 222222 −−−−−− −−−−−− )2222(2 53108 −−−− −−−− 12103.2 −× )11(h 16106 222 −−− +−− )22(2 406 −− −− 12108.1 −× )12(h 20181413107 222222 −−−−−− ++−−− )22(2 307 −− − 10105.2 −× )13(h 2018161312105 2222222 −−−−−−− −−−+++− )22(2 505 −− +− 10104.9 −× )14(h 18171311 2222 −−−− ++−− )22(2 2011 −− −− 12101 −× )15(h 201814131074 2222222 −−−−−−− +++++− )22(2 304 −− − 8101.1 −× )16(h 13986 2222 −−−− −−−− )222(2 3206 −−− −−− 10102.1 −× )17(h 15131063 22222 −−−−− ++++− )22(2 303 −− +− 9109.9 −× )18(h 20191613963 2222222 −−−−−−− −−−−+− )22(2 303 −− − 8106.2 −× )19(h 19181513861 2222222 −−−−−−− −−−−−− )22(2 501 −− − 7103.1 −×

Fig. 4. Implementation of the filter tap, )(0 nh . Fig. 5. PFP Implementation of the filter tap.

TABLE I Coefficients of the FIR filter in Example 1.



Fig. 6. Filter frequency responses in Example 1. Solid line: 10-bit PFP, Dotted line: 20-bit SOPOT (both responses coincide).

Fig. 7. Impulse response of the FIR subfilter in Example 1 using 20-bit SOPOT coefficients (marked with cross) and 10-bit PFP coefficients (marked with circle) obtained by the

proposed algorithm (both characteristics coincide).



)(nh 8-bit PFP Coefficients (shift: 3-bit, span: 5-bit)

Sensitivity)128( =M

)0(h 0 0 )1(h 72− 8106.9 −×)2(h 1187 222 −−− −−− 10106.3 −×)3(h 1197 222 −−− −−− 11105.6 −×)4(h 0 0 )5(h 11987 2222 −−−− +++ 11109.8 −×)6(h 76 22 −− + 11103.5 −×)7(h 86 22 −− + 10101.3 −×)8(h 0 0 )9(h 11876 2222 −−−− −−−− 11105.4 −×)10(h 65 22 −− −− 9102.7 −×)11(h 975 222 −−− −−− 9101.6 −×)12(h 0 0 )13(h 974 222 −−− ++ 9107.5 −×)14(h 53 22 −− + 9104.9 −×)15(h 8543 2222 −−−− +++ 8105.2 −×)16(h 22− 0

Mixer

2↓ )(1 zHMHz 72.30=sF BW=5 MHz

2↓ )(1 zH

15.36 MHz

W-CDMA signal

Fig. 8. IF block of the W-CDMA receiver.

TABLE II Coefficients of the filter in Example 2.



Example 1 Example 2 SOPOT (16-bit)

PFP (BSR)

PFP (ASR)

SOPOT (16-bit)

PFP (BSR)

PFP (ASR)

1599 1132 282 1233 896 230 Reduction 29% 82% 27.3% 81%

Implementation method Example 1 Example 2 Conventional SOPOT 107 85

Proposed PFP 63 45 Percentage Reduction 41% 47%

TABLE III Number of adders required to implement the filters in design examples.

TABLE IV Number of full adders required to implement the filters in design examples.

Fig. 9. Filter frequency responses in Example 2. Solid line: 8-bit PFP, Dotted line: 16-bit SOPOT

(Frequency scale indicated is 0 – 7.68 MHz).



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